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Electrical bio-impedance (EBI) describes the electrical properties biological materials exhibit as current flows through them. EBI is commonly measured by injecting a small sinusoidal alternating current (AC) into the tissue under study. The injected current induces an electrical field within the tissue and results in a measurable voltage drop across it. The AC version of Ohm's Law (Equation 1) relates the tissue's electrical bio-impedance Z to the ratio between the voltage V (i.e., the voltage drop across the tissue) and the injected current I.
                    Z        =                  V          I                                    Eq        .                                  ⁢        1            
Z is a complex quantity, since biomaterials not only oppose current flow, but also store electrical charge and phase-shift the voltage with respect to the current in the time-domain.
When current flows through a tissue, it passes through extracellular and intracellular fluids. These fluids are highly conductive as they contain salt ions that can easily be displaced by a potential difference. Conversely, the cells' lipid membranes are insulators. They act like capacitive plates, which prevent electrical charges from flowing through. Accordingly, the tissue's impedance reflects its chemical composition, membrane structures, and fluids distribution. For similar reasons, the specific cell types (blood, adipose, muscle, bone, etc.), the anatomic configuration (i.e., bone or muscle orientation and quantity), and the state of the cells (normal or osteoporotic bone, oedematous vs. normally hydrated tissue, etc.) affect measured impedance quantities.
Most tissues display dispersive characteristics, i.e. their impedance varies with the frequency of the applied current. A typical dispersion curve may be plotted as a Cole-Cole plot which superimposes impedance measurements from a range of frequencies on the complex plane. At low frequencies, the cells' membranes block the current. Thus, the impedance corresponds only to the extracellular resistance. As frequency increases, more current passes through the intracellular capacitive path, and the phase angle accumulates. At high frequencies, the intracellular capacitance becomes negligible. The impedance is once again purely resistive, dominated by the intracellular and extracellular fluid resistances connected in parallel. The frequency at which the tissue's reactance reaches a peak is known as the center frequency (fc).
A simple electrical circuit that may be used to model EBI response is (Ri+Cm))∥Re, that is, Re in parallel with the series combination of Ri and Cm. In this model, Ri is intracellular resistance; Re is extracellular resistance, and Cm is cell membrane capacitance. This model results in a perfect semicircle in a Cole-Cole plot, with the center of the circle on the resistance axis.
In real tissue, however, the cells' membrane is an imperfect capacitor. Moreover, the large variation in cell type, structure, and size causes a distribution of the cells' capacitive time constants. Cole and Cole showed that when capacitive time constant distribution is added to the above circuit model, the impedance is related to the frequency by:
                    Z        =                              R            ∞                    +                                    R              0                                      1              +                                                (                                      j                    ⁢                                                                                  ⁢                    ω                    ⁢                                                                                  ⁢                    τ                                    )                                                  1                  -                  α                                                                                        Eq        .                                  ⁢        2            whereR0=Re  Eq. 3R∞=Ri∥Re  Eq. 4τ=(Re+Ri)Cm  Eq. 5
In Equations 2-5 above, ω is angular frequency of electrical current in radians per second, Z is complex impedance (in ohms), R0 is resistance (in ohms) at zero frequency, R∞ is resistance (in ohms) at infinite frequency, τ is a time constant (in seconds) that is the inverse of the characteristic frequency ωc, and α is a dimensionless exponent that is proportional to depression angle, as discussed below.
This model preserves the circular shape in a Cole-Cole plot, but depresses the circle's center below the resistance axis. α has a value between 0 and 1, and is proportional to the angle to the depressed center.
Two types of electrode systems are commonly used to obtain EBI measurements. A two-electrode system uses the same pair of electrodes to inject current (IC) and pick up (PU) the tissue's response.
A four-electrode system uses different pairs for excitement and pick up. A pair of IC (inject current) electrodes drive current into tissue and a pair of PU (pick up) electrodes measure voltage response of the tissue. The type of electrodes in use (needle or skin surface, gel or dry, etc.), and their configuration around the tissue affect the sensed impedance almost as much as the electrical properties of the tissue.
The electrode type determines how the electrical conductor in the measurement leads interfaces with the ionic conductor in the biological tissue. As current flows, substance concentration may change near the electrodes' interface, adding bias impedance called electrode polarization. The skin-electrode contact introduces an additional resistive bias. The four-electrode system is a robust setup that reduces the influence of these factors. When voltage pick up is implemented with high-impedance differential amplifiers, such artifacts can be neglected.
The electrode configuration sets boundary values on the electrical fields that develop inside the tissue. Thus, it governs the fields' propagation and in effect, the relative contribution of internal tissue regions to the measured mutual impedance. For a four-electrode system, the measured impedance Z resulting from the variable conductivity σ within a volume conductor can be evaluated by:
                    Z        =                              ∫            V                    ⁢                                    1              σ                        ⁢            S            ⁢                                                  ⁢            d            ⁢                                                  ⁢            v                                              Eq        .                                  ⁢        6            
Sensitivity S is a scalar field, determining the contribution of a local conductivity change (Δσ) to the overall potential. It may be calculated from the dot product of two current density fields:S={right arrow over (JIC)}·{right arrow over (JPU)}  Eq. 7
{right arrow over (JIC)} represents the current density field generated by a unit current applied through the IC electrodes. {right arrow over (JPU)} is the reciprocal current density field that would have been generated had the same current been injected through the PU electrodes.
Depending on the angle of the two fields, there may be regions where the sensitivity is zero, positive, or negative. Hence, the tissue regions, in which impedance changes are measured, may effectively be manipulated by the electrode configuration. The measured mutual impedance may be indifferent to interchanges between the IC and PU electrodes.